Given a pair \((x_1,x_2)\)  and its latent representation \((z_1,z_2)\) , assume the existence of an oracle \(\mathcal {O}\) , that, acting on the latent vectors, tells us exactly which subspace \(\mathcal {S}_i\)  of \(\tilde{\mathcal {Z}}\)  red\(\mathcal {Z}\)  contains the difference in the input pair.
If \(x_2 = T_i (x_1)\)  for a fixed \(T_i\)  acting on \(\mathcal {M}_i\)  then  \(\mathcal {O}(z_1,z_2)= i\)  red\(\mathcal {O}(\tilde{z}_1,\tilde{z}_2)= i .\) 
The oracle \(\mathcal {O}\)  has been implemented in practice in [1]}, [2]} by allowing a higher degree of supervision, i.e. by incorporating labels into the sampled data pairs.
However, in real settings, typically we do not have access to labels, thus \(\mathcal {O}\)  has to be estimated.
